1. Chapter 4: Elementary Number Theory and Methods of Proof
Discrete Structures
Chapter 4: Elementary Number Theory and Methods of Proof
4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem
Be especially critical of any statement following the word “obviously”.
– Anna Pell Wheeler,
4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

2. Theorem 4.4.1 – The Quotient-Remainder Theorem
Given any integer n and positive integer d,  unique integers q and r s.t. n = dq + r and 0  r < d
4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

3. Definitions Given an integer n and a positive integer d,
n div d = the integer quotient obtained when n is divided by d.
n mod d = the nonnegative integer remainder obtained when n is divided by d.
Symbolically, if n and d are integers and d > 0, then
n div d = q and n mod d = r  n = dq + r
Where q and r are integers and 0  r < d.
4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

4. Example – pg. 189 # 8 & 9 Evaluate the expressions.
a. 50 div 7 b. 50 mod 7
a. 28 div 5 b. 28 mod 5
4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

5. Theorems Theorem 4.4.2 – The Parity Property
Any two consecutive integers have opposite parity.
Theorem 4.4.3
The square of any odd integer has the form 8m + 1 for some integer m.
4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

6. Definition
For any real number x, the absolute value of x, is denoted |x|, is defined as follows:
4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

7. Lemmas Lemma 4.4.4 For all real numbers r, -|r|  r  |r| Lemma 4.4.5
Lemma – The Triangle Inequality
For all real numbers x and y, |x + y|  |x| + |y|
4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

8. Example – pg. 189 # 24
Prove that for all integers m and n, if m mod 5 = 2 and n mod 5 = 1, then mn mod 5 = 2.
4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

9. Example – pg. 190 #36 Prove the following statement.
The product of any four consecutive integers is divisible by 8.
4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

10. Example – pg. 190 #42 Prove the following statement.
Every prime number except 2 and 3 has the form 6q + 1 or 6q + 5 for some integer q.
4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

11. Example – pg. 190 #45 Prove the following statement.
For all real numbers r and c with c  0, if c  r  c, then |r|  c.
4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

12. Example – pg. 190 # 49
If m, n, and d are integers, d > 0, and m mod d = n mod d, does it necessarily follow that m = n? That m – n is divisible by d? Prove your answers.
4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem